Related papers: M-systems and Cluster algebras
Clustering is an unsupervised machine learning methodology where unlabeled elements/objects are grouped together aiming to the construction of well-established clusters that their elements are classified according to their similarity. The…
In this paper we consider the class of l-bijective C-systems, i.e., C-systems for which the length function is a bijection. The main result of the paper is a construction of an isomorphism between two categories - the category of…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
Multicomponent systems are defined as chemical systems that require a quantum mechanical description of two or more different types of particles. Non-Born-Oppenheimer electron-nuclear interactions in molecules, electron-hole interactions in…
Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the…
We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…
Clustering methods with dimension reduction have been receiving considerable wide interest in statistics lately and a lot of methods to simultaneously perform clustering and dimension reduction have been proposed. This work presents a novel…
Several versions of approximate conjugacy for minimal dynamical systems are introduced. Relation between approximate conjugacy and corresponding crossed product $C^*$-algebras is discussed. For the Cantor minimal systems, a complete…
Multi-view clustering is an important and fundamental problem. Many multi-view subspace clustering methods have been proposed, and most of them assume that all views share a same coefficient matrix. However, the underlying information of…
The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two…
This is a survey article on some connections between cluster algebras and link invariants, written for the Notices of the AMS.
In recent years, a better understanding of the Monte Carlo method has provided us with many new techniques in different areas of statistical physics. Of particular interest are so called cluster methods, which exploit the considerable…
The clustering of categories in a multivariate categorical data set is investigated, where the problem separates into that of merging categories of the same variables (i.e., within-variable categories), and combining categories of different…
In 2010, Hernandez and Leclerc studied connections between representations of quantum affine algebras and cluster algebras. In 2019, Brito and Chari defined a family of modules over quantum affine algebras, called Hernandez-Leclerc modules.…
Every state on the algebra $M_n$ of complex nxn matrices restricts to a state on any matrix system. Whereas the restriction to a matrix system is generally not open, we prove that the restriction to every *-subalgebra of $M_n$ is open. This…
In this paper, we characterize all the finite dimensional algebras that are derived equivalent to an m-cluster tilted algebra of type A tilde. This generalizes a result of Bobonski and Buan [9].
Clustering is one of the most fundamental problems in data analysis and it has been studied extensively in the literature. Though many clustering algorithms have been proposed, clustering theories that justify the use of these clustering…
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
In this work we present a coupled-cluster theory for the propagation of multireference electronic systems initiating at general quantum mechanical states. Our formalism is based on the infinitesimal analysis of modified cluster operators,…
$Q$-systems are recursion relations satisfied by the characters of the restrictions of special finite-dimensional modules of quantum affine algebras. They can also be viewed as mutations in certain cluster algebras, which have a natural…